On supercompactness and the continuum function

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Given a cardinal κ that is λ-supercompact for some regular cardinal λ≥κ and assuming, we show that one can force the continuum function to agree with any function F:[κ,λ]∩→ satisfying ∀α,β∈(F) α<(F(α)) and α<β F(α)≤ F(β), while preserving the λ-supercompactness of κ from a hypothesis that is of the weakest possible consistency strength, namely, from the hypothesis that there is an elementary embedding j:V→ M with critical point κ such that Mλ M and j(κ)>F(λ). Our argument extends Woodin's technique of surgically modifying a generic filter to a new case: Woodin's key lemma applies when modifications are done on the range of j, whereas our argument uses a new key lemma to handle modifications done off of the range of j on the ghost coordinates. This work answers a question of Friedman and Honzik [FH2012]. We also discuss several related open questions.